The problem I'm having is, I believe, related to the fact that the Dropbox folders have different locations on my computers. On one computer, it's D:\Dropbox. On another, it's C:\Users\[my name]\Dropbox\cosmic.

I'm unable to open projects, often on either computer, getting an error along the lines of "Unable to read project /D/Dropbox/cosmic/Cosmic.sublime-project". I'm getting this error on the computer where Dropbox is on the C drive, so it's trying to look in the directory from the other computer. After getting this error, ST will sometimes open the last files I had open, sometimes not, but it won't load the folder into the sidebar, and I'll have weird behavior in ST until I click "Close Project" and restart it.

I never had any issues like this in Sublime Text 2, and used the exact same setup.

I have the same issue (also with relative project paths like your "."). I, too, sync the project files through Dropbox, and I, too, had no issues doing so in ST2.

As a temporary solution more practical than deleting the project and creating it again every time I switch computers, I have found that deleting the workspace file for the project (while ST3 is closed) allows the projects to be opened. However, this obviously means our workspace is preserved. I bet the issue is due to the workspace file have the "wrong" file paths (because we use it on different computers).